Category: mathematics

I work on the mathematics of sharing resources, which has led me to consider emotions such as envy, behaviour such as risk-taking and the best way to cut a cake.

Like, I suspect, many women, my wife enjoys eating dessert but not ordering it. I therefore dutifully order what I think she’ll like, cut it in half and invite her to choose a piece.

This is a sure-fire recipe for marital accord. Indeed, many mathematicians, economists, political scientists and others have studied this protocol and would agree. The protocol is known as the “cut-and-choose” procedure. I cut. You choose.

Cut-and-choose

Cut-and-choose is not limited to the dining table – it dates back to antiquity. It appears nearly 3,000 years ago in Hesiod’s poem Theogeny where Prometheus divides a cow and Zeus selects the part he prefers.

In more recent times, cut-and-choose has been enshrined in the UN’s 1982 Convention of the Law of the Sea where it was proposed as a mechanism to resolve disputes when dividing the seabed for mining.

To study the division of cake, cows and the seabed in a more formal way, various mathematical models have been developed. As with all models, these need to make a number of simplifying assumptions.

One typical assumption is that the people employing the cut-and-choose method are risk-averse. They won’t adopt a risky strategy that may give them less cake than a more conservative strategy.

With such assumptions in place, we can then prove what properties cake cutting procedures have and don’t have. For instance, cut-and-choose is envy free.

You won’t envy the cake I have, otherwise you would have taken this piece. And I won’t envy the piece you have, as the only risk-averse strategy is for me to cut the cake into two parts that I value equally.

On the other hand, the cutting of the cake is not totally equitable since the player who chooses can get cake that has more than half the total value for them.

With two players, it’s hard to do better than cut-and-choose. But I should record that my wife argues with me about this.

She believes it favours the second player since the first player inevitably can’t divide the cake perfectly and the second player can capitalise on this. This is the sort of assumption ignored in our mathematical models.

My wife might prefer the moving-knife procedure which doesn’t favour either player. A knife is moved over the cake, and either player calls “cut” when they are happy with the slice.

Again, this will divide the cake in such a way that neither player will envy the other (else they would have called “cut” themselves).

Three’s a crowd

Unfortunately, moving beyond two players increases the complexity of cutting cake significantly.

With two players, we needed just one cut to get to an envy free state. With three players, a complex series of five cuts of the cake might be needed. Of course, only two cuts are needed to get three slices.

The other three cuts are needed to remove any envy. And with four players, the problem explodes in our face.

An infinite number of cuts may be required to get to a situation where no one envies another’s cake. I’m sure there’s some moral here about too many cake cutters spoiling the dessert.

There are many interesting extensions of the problem. One such extension is to indivisible goods.

Suppose you have a bag of toys to divide between two children. How do you divide them fairly? As a twin myself, I know that the best solution is to ensure you buy two of everything.

It’s much more difficult when your great aunt gives you one Zhu Zhu pet, one Bratz doll and three Silly Bandz bracelets to share.

Online

More recently, I have been studying a version of the problem applicable to online settings. In such problems, not all players may be available all of the time. Consider, for instance, allocating time on a large telescope.

Astronomers will have different preferences for when to use the telescope depending on what objects are visible, the position of the sun, etcetera. How do we design a web-based reservation system so that astronomers can choose observation times that is fair to all?

We don’t want to insist all astronomers log in at the same time to decide an allocation. And we might have to start allocating time on the telescope now, before everyone has expressed their preferences. We can view this as a cake-cutting problem where the cake is made up of the time slots for observations.

The online nature of such cake-cutting problems poses some interesting new challenges.

How can we ensure that late-arriving players don’t envy cake already given to earlier players? The bad news is that we cannot now achieve even a simple property like envy freeness.

No procedure can guarantee situations where players don’t envy one another. But more relaxed properties are possible, such as not envying cake allocated whilst you are participating in the cutting of the cake.

Ham sandwich

There’s a brilliantly named piece of mathematics due to Arthur H. Stone and [John Tukey](http://www.morris.umn.edu/~sungurea/introstat/history/w98/Tukey.html, the Ham Sandwich Theorem which proves we can always cut a three-layered cake perfectly with a single cut.

Suppose we have three objects. Let’s call them “the top slice of bread”, “the ham filling” and “the bottom slice of bread”. Or if you prefer “the top layer” of the cake, “the middle layer” and “the bottom layer”.

The ham sandwich theorem proves a single slice can always perfectly bisect the three objects. Actually, the ham sandwich theorem works in any number of dimensions: any n objects in n-dimensional space can be simultaneously bisected by a single (n − 1) dimensional hyperplane.

So, in the case of the three-layered cake, n = 3, and the three-layered cake can be bisected (or cut) using a single, two-dimensional “hyperplane”. Such as, say, a knife.

Who would have thought that cutting cake would lead to higher dimensions of mathematics by way of a ham sandwich?

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Toby Walsh*

The diagram above (known as the tetractys) shows the first four triangular numbers (1, 3, 6, 10, …). Although there is a simple formula for calculating these numbers directly, t(n) = 1/2(n(n+1)), constructing them by these layered-triangle diagrams helps to show their geometric and recursive properties.

More generally, polygonal numbers arise from counting arrangements of dots in regular polygonal patterns. Larger polygons are built from smaller ones of the same type by adding additional layers of dots, called gnomons. Beginning with a single dot, k-sided polygons are built by adding gnomons consisting of k-2 segments, with each segment of the gnomon having one more dot than the segments of the previous layer. In this way, the nth gnomon consists of segments each n dots long, but with k-3 dots shared by adjoining segments (the corners).

This post describes how you can draw figures that illustrate the polygonal numbers and explore the polygonal numbers in general (triangular, square, pentagonal, hexagonal, etc.) using either TinkerPlots or Fathom. Both TinkerPlots and Fathom work well, but TinkerPlots creates nicer pictures, and allows for captions directly on the graph.
Without describing the details of how you create Fathom or TinkerPlot documents, here are the attributes that you will want to define in order to draw diagrams like the ones shown.

Required attributes
Create a slider k. This will allow you to set what kind of polygonal number you want to draw (k=3 gives triangular numbers, k=4 gives square numbers, etc.)
Define the following attributes:

n The number itself. This is a natural number beginning at 1 and continuing through the number of cases.

gnomon This states which “layer” or gnomon the number belongs in. It is calculated based on a number of other attributes.

g_index This is the position of the number within the gnonom – it ranges from 1 up until the next k-polygonal number is hit.

s_index Each gonom is broken up into sections or sides – what is the position within the side? Each side is of length equal to the gonom number. The first gonom has sides of length 1, the second has length 2, etc.

corner This keeps track of whether or not the number is a “corner” or not. This is based primarily on the s_index attribute.

c_index This keeps track of how many corners we have so far. There are only k-1 corners in a gnomon (the first number n=1 is the remaining corner). So, when we hit the last corner, we know we are at a polygonal number.

k_poly Records whether or not the number n is k-polygonal. It does this by checking to see if it is the last corner of a gnonom.
The attributes listed above are required for finding the position of each number within the figure; th following attributes are used in actually drawing the figures.

angle The base corner angle for the polygon is determined by k. This is the external angle for each corner.

current_angle We have to add to the base angle at each corner as we turn at each corner. This attribute is used to keep track of the total current angle.

dx This is the x-component of the unit direction vector that we are travelling in. Each new dot moves one dx over in the x-direction. It is given by the cosine of the current angle.

dy This is the y-component of the unit direction vector that we are travelling in. Each new dot moves one dy over in the y-direction. It is given by the sine of the current angle.

prev_g_1_x This is the x-coordinate of the first dot in the previous gonom layer. We need to know this because it will be the starting point for the next layer – each layer starts back at the “beginning” of the figure.

prev_g_1_y This is the y- coordinate of the first dot in the previous gonom layer.

x This is the x-coordinate of the current dot, calculated either from the previous dot or from the first dot in the previous layer.

y This is the y-coordinate of the current dot, calculated either from the previous dot or from the first dot in the previous layer.

caption Used to display the number on the plot (TinkerPlots only)
Below are the formulas for each attribute, written in “ascii” math. They are presented without a full explanation, in the hopes that if you try to implement this you will think about and explore each using the formulas and the descriptions above as a guide. Alternate methods for drawing the diagrams are possible, and you might find other formulas that achieve the same goals. Note that there are nested if() statements in several formulas.

n = caseIndex

gnomon = if(n=1){1, if(prev(k_poly)){prev(gnomon)+1, prev(gnomon)

g_index = if(n=1){ 1, if(prev(k_poly){1, prev(g_index) +1

s_index = if(n=1){ 1, if(prev(k_poly){1, if(prev(s_index) = gnomon){2,prev(s_index)

corner = (s_index=1) or (s_index=gnomon)

c_index= if(g_index=1){1, if(corner){prev(c_index)+1, prev(c_index)

k_poly = if(n=1){true, (c_index=k-1)

prev_g_1_x = if (n=1){0, if(g_index=2){prev(x), prev(prev_g_1_x)

prev_g_1_y = if(n=1){0, if(g_index=2){prev(y), prev(prev_g_1_y)

angle = pi-((k-2)*(pi/k))

current_angle = if(g_index =1) {pi-angle, if(pref(corner)){prev(current_angle)-angle, prev(current_angle)

dx = cos(current_angle)

dy = sin(current_angle)

x= if(n=1){0, if(g_index=1){prev_g_1_x +dx, prev(x) +dx

y =if(n=1){0, if(g_index=1){prev_g_1_y +dy, prev(y) +dy

caption = if(k_poly){n,””

To actually draw the diagrams, create a new plot with the x and y attributes as the horizontal and vertical axis, respectively. Add cases to the collection to populate the diagram. Optionally you can show connecting lines, and (in TinkerPlots) add a legend using the caption attribute.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

Exciting a brain region using electrical noise stimulation can help improve mathematical learning in those who struggle with the subject, according to a new study from the Universities of Surrey and Oxford, Loughborough University, and Radboud University in The Netherlands.

During this unique study, published in PLOS Biology, researchers investigated the impact of neurostimulation on learning. Despite the growing interest in this non-invasive technique, little is known about the neurophysiological changes induced and the effect it has on learning.

Researchers found that electrical noise stimulation over the frontal part of the brain improved the mathematical ability of people whose brain was less excited (by mathematics) before the application of stimulation. No improvement in mathematical scores was identified in those who had a high level of brain excitation during the initial assessment or in the placebo groups. Researchers believe that electrical noise stimulation acts on the sodium channels in the brain, interfering with the cell membrane of the neurons, which increases cortical excitability.

Professor Roi Cohen Kadosh, Professor of Cognitive Neuroscience and Head of the School of Psychology at the University of Surrey who led this project, said, “Learning is key to everything we do in life—from developing new skills, such as driving a car, to learning how to code. Our brains are constantly absorbing and acquiring new knowledge.

“Previously, we have shown that a person’s ability to learn is associated with neuronal excitation in their brains. What we wanted to discover in this case is if our novel stimulation protocol could boost, in other words excite, this activity and improve mathematical skills.”

For the study, 102 participants were recruited, and their mathematical skills were assessed through a series of multiplication problems. Participants were then split into four groups including a learning group exposed to high-frequency random electrical noise stimulation and an overlearning group in which participants practiced the multiplication beyond the point of mastery with high-frequency random electrical noise stimulation.

The remaining two groups consisted of a learning and overlearning group but they were exposed to a sham (i.e., placebo) condition, an experience akin to real stimulation without applying significant electrical currents. EEG recordings were taken at the beginning and at the end of the stimulation to measure brain activity.

Dr. Nienke van Bueren, from Radboud University, who led this work under Professor Cohen Kadosh’s supervision, said, “These findings highlight that individuals with lower brain excitability may be more receptive to noise stimulation, leading to enhanced learning outcomes, while those with high brain excitability might not experience the same benefits in their mathematical abilities.”

Professor Cohen Kadosh adds, “What we have found is how this promising neurostimulation works and under which conditions the stimulation protocol is most effective. This discovery could not only pave the way for a more tailored approach in a person’s learning journey but also shed light on the optimal timing and duration of its application.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to University of Surrey

I am often asked the question, “How do you do it?!” Now while I don’t think my note-taking strategy is particularly special, I am happy to share! I’ll preface the information by stating what you probably already know: I LOVE to write.* I am a very visual learner and often need to go through the physical act of writing things down in order for information to “stick.” So while some people think aloud (or quietly),

I think on paper.

My study habits, then, are built on this fact. Of course not everyone learns in this way, so this post is not intended to be a how-to guide. It’s just a here’s-what-I-do guide.

With that said, below is a step-by-step process I tried to follow during my final years of undergrad and first two years of grad school.**

‍Step 1

Read the appropriate chapter/section in the book before class

I am an “active reader,” so my books have tons of scribbles, underlines, questions, and “aha” moments written on the pages. I like to write while I read because it gives me time to pause and think about the material. For me, reading a mathematical text is not like reading a novel. It often takes me a long time just to understand a single paragraph! Or a single sentence. I also like to mark things that I don’t understand so I’ll know what to look for in the upcoming lecture.

STEP 2

Attend lecture and take notes

This step is pretty self-explanatory, but I will mention this: I write down much more than what is written on the chalkboard (or whiteboard). In fact, a good portion of my in-class notes consists of what the professor has said but hasn’t written.

‍My arsenal

‍STEP 3

Rewrite lecture notes at home

My in-class notes are often an incomprehensible mess of frantically-scribbled hieroglyphs. So when I go home, I like to rewrite everything in a more organized fashion. This gives the information time to simmer and marinate in my brain. I’m able to ponder each statement at my own pace, fill in any gaps, and/or work through any exercises the professor might have suggested. I’ll also refer back to the textbook as needed.

Sometimes while rewriting these notes, I’ll copy things word-for-word (either from the lecture, the textbook, or both), especially if the material is very new or very dense. Although this can be redundant, it helps me slow down and lets me think about what the ideas really mean. Other times I’ll just rewrite things in my own words in a way that makes sense to me.

A semester’s worth of notes!

 

As for the content itself, my notes usually follow a “definition then theorem then proof” outline, simply because that’s how material is often presented in the lecture. But sometimes it’s hard to see the forest for the trees (i.e. it’s easy to get lost in the details), so I’ll occasionally write “PAUSE!” or “KEY IDEA!” in the middle of the page. I’ll then take the time to write a mini exposition that summarizes the main idea of the previous pages. I’ve found this to be especially helpful when looking back at my notes after several months (or years) have gone by. I may not have time to read all the details/calculations, so it’s nice to glance at a summary for a quick refresher.

And every now and then, I’ll rewrite my rewritten notes in the form of a SaiBlog post! Many of my earlier posts here at Math3ma were “aha” moments that are now engrained in my brain because I took the time to SaiBlog about them.

STEP 4

Do homework problems

Once upon a time, I used to think the following:

How can I do problems if I haven’t spent a bajillion hours learning the theory first?

But now I believe there’s something to be said for the converse: 

How can I understand the theory if I haven’t done a bajillion examples first?

In other words, taking good notes and understanding theory is one thing, but putting that theory into practice is a completely different beast. As a wise person once said, “The only way to learn math is to DO math.” So although I’ve listed “do homework problems” as the last step, I think it’s really first in terms of priority.

Typically, then, I’ll make a short to-do list (which includes homework assignments along with other study-related duties) each morning. And I’ll give myself a time limit for each task. For example, something like “geometry HW, 3 hours” might appear on my list, whereas “do geometry today” will not. Setting a time gives me a goal to reach for which helps me stay focused. And I may be tricking my brain here, but a specific, three-hour assignment sounds much less daunting than an unspecified, all-day task. (Of course, my lists always contain multiple items that take several hours each, but as the old adage goes, “How do you eat an elephant? One bite at a time.”)

By the way, in my first two years of grad school I often worked with my classmates on homework problems. I didn’t do this in college, but in grad school I’ve found it tricky to digest all the material alone – there’s just too much of it! So typically I’d first attempt exercises on my own, then meet up with a classmate or two to discuss our ideas and solutions and perhaps attend office hours with any questions.

As far as storage goes, I have a huge binder that contains all of my rewritten notes*** from my first and second year classes. (I use sheet protectors to keep them organized according to subject.) On the other hard, I use a paper tray like this one to store my lecture notes while the semester is in progress. Once classes are over, I’ll scan and save them to an external hard drive. I’ve also scanned and saved all my homework assignments.

Well, I think that’s about it! As I mentioned earlier, these steps were only my ideal plan. I often couldn’t apply them to every class — there’s just not enough time! — so I’d only do it for my more difficult courses. And even then, there might not be enough time for steps 1 and 3, and I’d have to start working on homework right after a lecture.

But as my advisor recently told me,”It’s okay to not know everything.” Indeed, I think the main thing is to just do something. Anything. As much as you can. And as time goes on, you realize you really are learning something, even if it doesn’t feel like it at the time.

Alright, friends, I think that’s all I have to share. I hope it was somewhat informative. If you have any questions, don’t hesitate to leave it in a comment below!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

Maths tells us the best way to cover a surface with copies of a shape – even when it comes to jigsaw puzzles, says Katie Steckles.

WHAT do a bathroom wall, a honeycomb and a jigsaw puzzle have in common? Obviously, the answer is mathematics.

If you are trying to cover a surface with copies of a shape – say, for example, you are tiling a bathroom – you ideally want a shape like a square or rectangle. They will cover the whole surface with no gaps, which is why these boring shapes get used as wall tiles so often.

But if your shapes don’t fit together exactly, you can still try to get the best coverage possible by arranging them in an efficient way.

Imagine trying to cover a surface with circular coins. The roundness of the circles means there will be gaps between them. For example, we could use a square grid, placing the coins on the intersections. This will cover about 78.5 per cent of the area.

But this isn’t the most efficient way: in 1773, mathematician Joseph-Louis Lagrange showed that the optimal arrangement of circles involves a hexagonal grid, like the cells in a regular honeycomb – neat rows where each circle sits nestled between the two below it.

In this situation, the circles will cover around 90.7 per cent of the space, which is the best you can achieve with this shape. If you ever need to cover a surface with same-size circles, or pack identical round things into a tray, the hexagon arrangement is the way to go.

But this isn’t just useful knowledge if you are a bee: a recent research paper used this hexagonal arrangement to figure out the optimal size table for working a jigsaw puzzle. The researchers calculated how much space would be needed to lay out the pieces of an unsolved jigsaw puzzle, relative to the solved version. Puzzle pieces aren’t circular, but they can be in any orientation and the tabs sticking out stop them from moving closer together, so each takes up a theoretically circular space on the table.

By comparing the size of the central rectangular section of the jigsaw piece to the area it would take up in the hexagonal arrangement, the paper concluded that an unsolved puzzle takes up around 1.73 times as much space.

This is the square root of three (√3), a number with close connections to the regular hexagon – one with a side length of 1 will have a height of √3. Consequently, there is also a √3 in the formula for the hexagon’s area, which is 3/2 × √3 × s², where s is the length of a side. This is partly why it pops out, after some fortuitous cancellation, as the answer here.

So if you know the dimensions of a completed jigsaw puzzle, you can figure out what size table you need to lay out all the pieces: multiply the width and height, then multiply that by 1.73. For this ingenious insight, we can thank the bees.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Katie Steckles*

There are plenty of probability problems that have counter-intuitive solutions. Problems like these, and how they can undermine common sense, are among the best reasons for looking at probability theory. One set of humbling questions is the family of Monty Hall problems. Another are those related to conditional probability; nice examples of these are problems that involve medical tests that give ‘false positive’ results.

Simulation is a way of exploring these problems that reveals more than mere theoretical probability calculations do. The structure of the simulation can reflect interesting aspects of the structure of the original problem, and the results reveal a variability that is not apparent when you simply calculate the theoretical probabilities on their own.

This post shows an example of a ‘false positive’ probability problem and a Fathom simulation for it. This problem was adapted from one found in the 4th edition of Ross’s A First Course in Probability (p.75):

A laboratory blood test is 95 percent effective in detecting a certain disease when it is, in fact, present. However, the test also yields a “false positive” result for one percent of healthy persons. (That is, if a healthy person is tested, there is a 0.01 probability that they will test positive for the disease, even though they don’t really have it.) If 0.5 percent of the population actually has the disease, what is the probability that a person who has a positive test result actually has the disease?

Here are the attribute definitions that you could use to build a Fathom simulation for this problem:

The attributes are enough to run the simulation, but it is better to also add the following measures:

To run the simulation you can add new cases (try ~500). Using a measures collection, you can re-run this experiment hundreds of times (collecting a 100 measures re-runs the 500 person experiment 100 times).

If you are calculating the theoretical probability by hand, it helps to write down all of the probabilities (fill in the blanks…):

It also helps to visualize the probabilities in a tree diagram:

The outer tips of the tree are filled in using the multiplicative rule for conditional probabilities:

One nice thing about doing this is that you can see how the tree diagram used to calculate the theoretical probabilities is structured in the same way as the “if” statements in the simulation.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

Knot theory is linked to many other branches of science, including those that tell us about the cosmos.

The mathematical study of knots started with a mistake. In the 1800s, mathematician and physicist William Thomson, also known as Lord Kelvin, suggested that the elemental building blocks of matter were knotted vortices in the ether: invisible microscopic currents in the background material of the universe. His theory dropped by the wayside fairly quickly, but this first attempt to classify how curves could be knotted grew into the modern mathematical field of knot theory. Today, knot theory is not only connected to many branches of theoretical mathematics but also to other parts of science, like physics and molecular biology. It’s not obvious what your shoelace has to do with the shape of the universe, but the two may be more closely related than you think.

As it turns out, a tangled necklace offers a better model of a knot than a shoelace: to a mathematician, a knot is a loop in three-dimensional space rather than a string with loose ends. Just as a physical loop of string can stretch and twist and rotate, so can a mathematical knot – these loops are floppy rather than fixed. If we studied strings with free ends, they could wiggle around and untie themselves, but a loop stays knotted unless it’s cut.

Most questions in knot theory come in two varieties: sorting knots into classes and using knots to study other mathematical objects. I’ll try to give a flavour of both, starting with the simplest possible example: the unknot.

Draw a circle on a piece of paper. Congratulations, you’ve just constructed an unknot! This is the name for any loop in three-dimensional space that is the boundary of a disc. When you draw a circle on a piece of paper, you can see this disc as the space inside the circle, and your curve continues to be an unknot if you crumple the paper up, toss it through the air, flatten it out and then do some origami. As long as the disc is intact, no matter how distorted, the boundary is always an unknot.

Things get more interesting when you start with just the curve. How can you tell if it’s an unknot? There may secretly be a disc that can fill in the loop, but with no limits on how deformed the disc could be, it’s not clear how you can figure this out.

Two unknots

It turns out that this question is both hard and important: the first step in studying complicated objects is distinguishing them from simple ones. It’s also a question that gets answered inside certain bacterial cells each time they replicate. In the nuclei of these cells, the DNA forms a loop, rather than a strand with loose ends, and sometimes these loops end up knotted. However, the DNA can replicate only when the loop is an unknot, so the basic life processes of the cell require a process for turning a potentially complicated loop into an unknotted one.

A class of proteins called topoisomerases unknot tangled loops of DNA by cutting a strand, moving the free ends and then reattaching them. In a mathematical context, this operation is called a “crossing change”, and it’s known that any loop can be turned into the unknot by some number of crossing changes. However, there’s a puzzle in this process, since random crossing changes are unlikely to simplify a knot. Each topoisomerase operates locally, but collectively they’re able to reliably unknot the DNA for replication. Topoisomerases were discovered more than 50 years ago, but biologists are still studying how they unknot DNA so effectively.

When mathematicians want to identify a knot, they don’t turn to a protein to unknot it for them.  Instead, they rely on invariants, mathematical objects associated with knots. Some invariants are familiar things like numbers, while others are elaborate algebraic structures. The best invariants have two properties: they’re practical to compute, given the input of a specific knot, and they distinguish many different classes of knots from each other. It’s easy to define an invariant with only one of these properties, but a computable and effective knot invariant is a rare find.

The modern era of knot theory began with the introduction of an invariant called the Jones Polynomial in the 1980s. Vaughan Jones was studying statistical mechanics when he discovered a process that assigns a polynomial – a type of simple algebraic expression – to any knot. The method he used was technical, but the essential feature is that no amount of wiggling, stretching or twisting changes the output. The Jones Polynomial of an unknot is 1, no matter how complicated the associated disc might be.

Jones’s discovery caught the attention of other researchers, who found simpler techniques for computing the same polynomial. The result was an invariant that satisfies both the conditions listed above: the Jones Polynomial can be computed from a drawing of a knot on paper, and many thousands of knots can be distinguished by the fact that they have different Jones Polynomials.

However, there are still many things we don’t know about the Jones Polynomial, and one of the most tantalising questions is which knots it can detect. Most invariants distinguish some knots while lumping others together, and we say an invariant detects a knot if all the examples sharing a certain value are actually deformations of each other. There are certainly pairs of distinct knots with the same Jones Polynomial, but after decades of study, we still don’t know whether any knot besides the unknot has the polynomial 1. With computer assistance, experts have examined nearly 60 trillion examples of distinct knots without finding any new knots whose Jones Polynomials equal 1.

The Jones Polynomial has applications beyond knot detection. To see this, let’s return to the definition of an unknot as a loop that bounds a disc. In fact, every knot is the boundary of some surface – what distinguishes an unknot is that this surface is particularly simple. There’s a precise way to rank the complexity of surfaces, and we can use this to rank the complexity of knots. In this classification, the simplest knot is the unknot, and the second simplest is the trefoil, which is shown below.

Trefoil knot

To build a surface with a trefoil boundary, start with a strip of paper. Twist it three times and then glue the ends together. This is more complicated than a disc, but still pretty simple. It also gives us a new question to investigate: given an arbitrary knot, where does it fit in the ranking of knot complexity? What’s the simplest surface it can bound? Starting with a curve and then hunting for a surface may seem backwards, but in some settings, the Jones Polynomial answers this question: the coefficients of the knot polynomial can be used to estimate the complexity of the surfaces it bounds.

Joan Licata

Knots also help us classify other mathematical objects. We can visually distinguish the two-dimensional surface of sphere from the surface a torus (the shape of a ring donut), but an ant walking on one of these surfaces might need knot theory to tell them apart. On the surface of a torus, there are loops that can’t be pulled any tighter, while any loop lying on a sphere can contract to a point.

We live inside a universe of three physical dimensions, so like the ant on a surface, we lack a bird’s eye view that could help us identify its global shape. However, we can ask the analogous question: can each loop we encounter shrink without breaking, or is there a shortest representative? Mathematicians can classify three-dimensional spaces by the existence of the shortest knots they contain. Presently, we don’t know if some knots twisting through the universe are unfathomably long or if every knot can be made as small as one of Lord Kelvin’s knotted vortices.

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*Credit for article given to Joan Licata*

Whether you are sharing a cake or a coastline, maths can help make sure everyone is happy with their cut, says Katie Steckles.

One big challenge in life is dividing things fairly. From sharing a tasty snack to allocating resources between nations, having a strategy to divvy things up equitably will make everyone a little happier.

But it gets complicated when the thing you are dividing isn’t an indistinguishable substance: maybe the cake you are sharing has a cherry on top, and the piece with the cherry (or the area of coastline with good fish stocks) is more desirable. Luckily, maths – specifically game theory, which deals with strategy and decision-making when people interact – has some ideas.

When splitting between two parties, you might know a simple rule, proven to be mathematically optimal: I cut, you choose. One person divides the cake (or whatever it is) and the other gets to pick which piece they prefer.

Since the person cutting the cake doesn’t choose which piece they get, they are incentivised to cut the cake fairly. Then when the other person chooses, everyone is satisfied – the cutter would be equally happy with either piece, and the chooser gets their favourite of the two options.

This results in what is called an envy-free allocation – neither participant can claim they would rather have the other person’s share. This also takes care of the problem of non-homogeneous objects: if some parts of the cake are more desirable, the cutter can position their cut so the two pieces are equal in value to them.

What if there are more people? It is more complicated, but still possible, to produce an envy-free allocation with several so-called fair-sharing algorithms.

Let’s say Ali, Blake and Chris are sharing a cake three ways. Ali cuts the cake into three pieces, equal in value to her. Then Blake judges if there are at least two pieces he would be happy with. If Blake says yes, Chris chooses a piece (happily, since he gets free choice); Blake chooses next, pleased to get one of the two pieces he liked, followed by Ali, who would be satisfied with any of the pieces. If Blake doesn’t think Ali’s split was equitable, Chris looks to see if there are two pieces he would take. If yes, Blake picks first, then Chris, then Ali.

If both Blake and Chris reject Ali’s initial chop, then there must be at least one piece they both thought was no good. This piece goes to Ali – who is still happy, because she thought the pieces were all fine – and the remaining two pieces get smooshed back together (that is a mathematical term) to create one piece of cake for Blake and Chris to perform “I cut, you choose” on.

While this seems long-winded, it ensures mathematically optimal sharing – and while it does get even more complicated, it can be extended to larger groups. So whether you are sharing a treat or a divorce settlement, maths can help prevent arguments.

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*Credit for article given to Katie Steckles*

Buying a specific set of 27 tickets for the UK National Lottery will mathematically guarantee that you win something.

Buying 27 tickets ensures a win in the UK National Lottery

You can guarantee a win in every draw of the UK National Lottery by buying just 27 tickets, say a pair of mathematicians – but you won’t necessarily make a profit.

While there are many variations of lottery in the UK, players in the standard “Lotto” choose six numbers from 1 to 59, paying £2 per ticket. Six numbers are randomly drawn and prizes are awarded for tickets matching two or more.

David Cushing and David Stewart at the University of Manchester, UK, claim that despite there being 45,057,474 combinations of draws, it is possible to guarantee a win with just 27 specific tickets. They say this is the optimal number, as the same can’t be guaranteed with 26.

The proof of their idea relies on a mathematical field called finite geometry and involves placing each of the numbers from 1 to 59 in pairs or triplets on a point within one of five geometrical shapes, then using these to generate lottery tickets based on the lines within the shapes. The five shapes offer 27 such lines, meaning that 27 tickets bought using those numbers, at a cost of £54, will hit every possible winning combination of two numbers.

The 27 tickets that guarantee a win on the UK National Lottery

Their research yielded a specific list of 27 tickets (see above), but they say subsequent work has shown that there are two other combinations of 27 tickets that will also guarantee a win.

“We’ve been thinking about this problem for a few months. I can’t really explain the thought process behind it,” says Cushing. “I was on a train to Manchester and saw this [shape] and that’s the best logical [explanation] I can give.”

Looking at the winning numbers from the 21 June Lotto draw, the pair found their method would have won £1810. But the same numbers played on 1 July would have matched just two balls on three of the tickets – still a technical win, but giving a prize of just three “lucky dip” tries on a subsequent lottery, each of which came to nothing.

Stewart says proving that 27 tickets could guarantee a win was the easiest part of the research, while proving it is impossible to guarantee a win with 26 was far trickier. He estimates that the number of calculations needed to verify that would be 10¹⁶⁵, far more than the number of atoms in the universe. “There’d be absolutely no way to brute force this,” he says.

The solution was a computer programming language called Prolog, developed in France in 1971, which Stewart says is the “hero of the story”. Unlike traditional computer languages where a coder sets out precisely what a machine should do, step by step, Prolog instead takes a list of known facts surrounding a problem and works on its own to deduce whether or not a solution is possible. It takes these facts and builds on them or combines them in order to slowly understand the problem and whittle down the array of possible solutions.

“You end up with very, very elegant-looking programs,” says Stewart. “But they are quite temperamental.”

Cushing says the research shouldn’t be taken as a reason to gamble more, particularly as it doesn’t guarantee a profit, but hopes instead that it encourages other researchers to delve into using Prolog on thorny mathematical problems.

A spokesperson from Camelot, the company that operates the lottery, told New Scientist that the paper made for “interesting reading”.

“Our approach has always been to have lots of people playing a little, with players individually spending small amounts on our games,” they say. “It’s also important to bear in mind that, ultimately, Lotto is a lottery. Like all other National Lottery draw-based games, all of the winning Lotto numbers are chosen at random – any one number has the same and equal chance of being drawn as any other, and every line of numbers entered into a draw has the same and equal chance of winning as any other.”

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*Credit for article given to Matthew Sparkes*

Knot patterns (which might not actually represent knots, but just braids or plaits), like the one shown here, are common in Celtic design.

One way to create knot patterns is to use a set of tiles that are put together following a few basic rules. This is not, as far as I know, a standard way to create these patterns – the book by Aidan Meehan, provides a technique for drawing the patterns by hand, rather than laying them out as tiles.

A wide range of knot patterns, including the one shown above, can be created using the set of six tiles shown below. These were constructed using Geometer’s Sketchpad.

The rules for putting these together are reasonably clear, but can be formalized. You can even come up with a notation for the tiles and formal rules for assembling them into knot patterns.

When laying down the tiles, you have the option of rotating them – some of the tiles don’t alter with rotation, some of them can be placed in two ways, some in four.

A nice property of this set is that you can’t paint yourself into a corner when using it – you can always find a tile that will compose with the pattern that you have started. This property means that this set of tiles is closed, in a certain sense.

A wider range of patterns can be created if you add tiles like the ones below to your set.

 

Unfortunately, when you include these tiles, your set is no longer closed. An open question (at least as far as I know) is – how many more tiles would you need to add in order to create a closed set of tiles that includes these? Also, what would these tiles look like? Do they lead to “reasonable” looking knots?

The knot pattern below (better described as a woven set of three links) was made using tiles including the ones from the “extended” set above.

 

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*Credit for article given to dan.mackinnon*